Download universal set

Section 2.1
Set Concepts
2.1-1
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What You Will Learn
Equality of sets
Application of sets
Infinite sets
2.1-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Set
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2.1-3
A set is a collection of objects, which
are called elements or members of the
set.
Three methods of indicating a set:
Description
Roster form
Set-builder notation
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Well-defined Set
A set is well defined if its contents can
be clearly defined.
Example:
The set of U.S. presidents is a well
defined set. Its contents, the
presidents, can be named.
2.1-4
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Example 1: Description of Sets
Write a description of the set containing
the elements Monday, Tuesday,
Wednesday, Thursday, Friday, Saturday,
Sunday.
2.1-5
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Example 1: Description of Sets
Solution
The set is the days of the week.
2.1-6
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Roster Form
Listing the elements of a set inside a pair
of braces, { }, is called roster form.
Example
{1, 2, 3,} is the notation for the set
whose elements are 1, 2, and 3.
(1, 2, 3,) and [1, 2, 3] are not sets.
2.1-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Naming of Sets
Sets are generally named with capital
letters.
Definition: Natural Numbers
The set of natural numbers or counting
numbers is N.
N = {1, 2, 3, 4, 5, …}
2.1-8
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Example 2: Roster Form of Sets
Express the following in roster form.
a) Set A is the set of natural
numbers less than 6.
Solution:
a) A = {1, 2, 3, 4, 5}
2.1-9
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Roster Form of Sets
Express the following in roster form.
b) Set B is the set of natural numbers
less than or equal to 80.
Solution:
b) B = {1, 2, 3, 4, …, 80}
2.1-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Roster Form of Sets
Express the following in roster form.
c) Set P is the set of planets in Earth’s
solar system.
Solution:
c) P = {Mercury, Venus, Earth, Mars,
Jupiter, Saturn, Uranus, Neptune}
2.1-11
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Set Symbols
•
The symbol ∈, read “is an element of,”
is used to indicate membership in a
set.
•
The symbol ∉ means “is not an
element of.”
2.1-12
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Set-Builder Notation
(or Set-Generator Notation)
2.1-13
•
A formal statement that describes
the members of a set is written
between the braces.
•
A variable may represent any one of
the members of the set.
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 4: Using Set-Builder
Notation
a) Write set B = {1, 2, 3, 4, 5} in
set-builder notation.
b) Write in words, how you would
read set B in set-builder notation.
2.1-14
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Example 4: Using Set-Builder
Notation
Solution
a)


or
B  x x N and x  5
B  x x N and x  6
b) The set of all x such that x is a
natural number and x is less than 6.
2.1-15
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Example 6: Set-Builder Notation
to Roster Form


Write set A  x x N and 2  x  8
in roster form.
Solution
A = {2, 3, 4, 5, 6, 7}
2.1-16
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Finite Set
A set that contains no elements or the
number of elements in the set is a
natural number.
Example:
Set B = {2, 4, 6, 8, 10} is a finite set
because the number of elements in the
set is 5, and 5 is a natural number.
2.1-17
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Infinite Set
A set that is not finite is said to be
infinite.
• The set of counting numbers is an
example of an infinite set.
•
2.1-18
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Equal Sets
Set A is equal to set B, symbolized
by A = B, if and only if set A and set
B contain exactly the same
members.
Example: { 1, 2, 3 } = { 3, 1, 2 }
2.1-19
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Cardinal Number
The cardinal number of set A,
symbolized n(A), is the number of
elements in set A.
Example:
A = { 1, 2, 3 } and
B = {England, Brazil, Japan}
have cardinal number 3,
n(A) = 3 and n(B) = 3
2.1-20
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Equivalent Sets
Set A is equivalent to set B if and only
if n(A) = n(B).
Example:
D={ a, b, c }; E={apple, orange, pear}
n(D) = n(E) = 3
So set A is equivalent to set B.
2.1-21
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Equivalent Sets - Equal Sets
Any sets that are equal must also be
equivalent.
• Not all sets that are equivalent are
equal.
Example:
D ={ a, b, c }; E ={apple, orange, pear}
n(D) = n(E) = 3; so set A is equivalent
to set B, but the sets are NOT equal
•
2.1-22
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One-to-one Correspondence
Set A and set B can be placed in oneto-one correspondence if every
element of set A can be matched with
exactly one element of set B and every
element of set B can be matched with
exactly one element of set A.
2.1-23
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One-to-one Correspondence
Consider set S states, and set C, state
capitals.
S = {North Carolina, Georgia, South
Carolina, Florida}
C = {Columbia, Raleigh, Tallahassee,
Atlanta}
Two different one-to-one
correspondences for sets S and C are:
2.1-24
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One-to-one Correspondence
S = {No Carolina, Georgia, So Carolina, Florida}
C = {Columbia, Raleigh, Tallahassee, Atlanta}
S = {No Carolina, Georgia, So Carolina, Florida}
C = {Columbia, Raleigh, Tallahassee, Atlanta}
2.1-25
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One-to-one Correspondence
Other one-to-one correspondences
between sets S and C are possible.
Do you know which capital goes with
which state?
2.1-26
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Null or Empty Set
The set that contains no elements is
called the empty set or null set and
is symbolized by
  or .
2.1-27
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Null or Empty Set
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•
2.1-28
Note that {∅} is not the empty set.
This set contains the element ∅ and
has a cardinality of 1.
The set {0} is also not the empty set
because it contains the element 0. It
has a cardinality of 1.
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Universal Set
•
•
2.1-29
The universal set, symbolized by U,
contains all of the elements for any
specific discussion.
When the universal set is given, only
the elements in the universal set may
be considered when working with the
problem.
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Universal Set
Example
If the universal set is defined as
U = {1, 2, 3, 4, ,…,10}, then only the
natural numbers 1 through 10 may be
used in that problem.
2.1-30
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